DANGLING OBJECT?

1995 Buick Century

There is something dangling on the bottom side of my car. It is about the exact center of the car. It is a cylinder about the size of a tomato paste can on a bracket. What is this? Is it important? I have no idea if this is in the right category as I don't know what this is.

That is a weight that attaches at the rear of the transmission to absorb vibration and it appears to have rusted off. You need to get it to a shop and see what it's going to take to get it attached solidly again.

Based on the presence of rust, that appears to be a weight added to the exhaust system to dampen vibration. If I'm right, it isn't important to the operation of the car, just comfort, assuming you will even feel any vibration.